JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (3): 77-84.doi: 10.6040/j.issn.1671-9352.0.2022.304

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A mollification method with a posteriori parameter selection for solving the Cauchy problem of the Helmholtz equation

LI Zhen-ping1,2, YU Ya-hui1*   

  1. 1. Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, Henan, China;
    2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2023-03-02

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Abstract: In order to restore the stability of the solution, a mollification regularization method with a posteriori selection of regularization parameter based on Gaussian kernel function is proposed. The stability error estimate between the exact solution and its approximation solution is obtained. Numerical experiments are carried out to verify the effectiveness of the proposed method.

Key words: Cauchy problem for the Helmholtz equation, mollification method, posteriori parameter selection, error estimate

CLC Number: 

  • O241.82
[1] REGINSKA T, REGINSKI K. Approximate solution of a Cauchy problem for the Helmholtz equation[J]. Inverse Probl, 2006, 22(3):975-989.
[2] FENG Xiaoli, FU Chuli, CHENG Hao. A regularization method for solving the Cauchy problem for the Helmholtz equation[J]. Appl Math Model, 2011, 35(7):3301-3315.
[3] XIONG Xiangtuan, FU Chuli. Two approximate methods of a Cauchy problem for the Helmholtz equation[J]. Comput Appl Math, 2007, 26(2):285-307.
[4] ZHANG H W, QIN H H, WEI T. A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation[J]. Int J Comput Math, 2011, 88(4):839-850.
[5] QIN H H, WEI T. Two regularization methods for the Cauchy problems of the Helmholtz equation[J]. Appl Math Model, 2010, 34(4):947-967.
[6] REGINSKA T. Regularization methods for a mathematical model of laser beams[J]. Eur J Math Comput Appl, 2014, 1(2):39-49.
[7] ZHANG Yuanxiang, FU Chuli, DENG Zhiliang. An a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations[J]. Inverse Prob Sci Eng, 2013, 21(7):1151-1168.
[8] FU Chuli, FENG Xiaoli, QIAN Zhi. The Fourier regularization for solving the Cauchy problem for the Helmholtz equation[J]. Appl Numer Math, 2009, 59(10):2625-2640
[9] QIAN Ailin, XIONG Xiangtuan, WU Yujiang. On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation[J]. J Comput Appl Math, 2010, 233(8):1969-1979.
[10] XIONG Xiangtuan. A regularization method for a Cauchy problem of the Helmholtz equation[J]. J Comput Appl Math, 2010, 233(8):1723-1732.
[11] HE Shangqin, FENG Xiufang. A mollification regularization method with the Dirichlet kernel for two Cauchy problems of three-dimensional Helmholtz equation[J]. Int J Comput Math, 2020, 97(11):2320-2336.
[12] HE Shangqin, DI Congna, YANG Li. The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundary[J]. Appl Numer Math, 2021, 160:422-435.
[13] LI Zhenping, XU Chao, LAN Man, et al. A mollification method for a Cauchy problem for the Helmholtz equation[J]. Int J Comput Math, 2018, 95(11):2256-2268.
[14] QIAN Zhi, FENG Xiaoli. A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation[J]. Appl Anal, 2017, 96(10):1656-1668.
[15] DINH N H. A mollification method for ill-posed problems[J]. Numer Math, 1994, 68(4):469-506.
[16] DINH N H, DUC N. Stability results for the heat equation backward in time[J]. J Math Anal Appl, 2009, 353(2):627-641.
[17] MURIO D A. The mollification method and the numerical solution of ill-posed problems[M]. New York: John Wiley and Sons Inc, 1993: 60-130.
[18] YANG Fan, FU Chuli. A mollification regularization method for the inverse spatial-dependent heat source problem[J]. J Comput Appl Math, 2014, 255:555-567.
[19] 邓志亮. 两类不适定问题的正则化方法研究[D]. 兰州: 兰州大学, 2010. DENG Zhiliang. The study of the regularization methods for two classes of ill-posed problems[D]. Lanzhou: Lanzhou University, 2010.
[20] 丁凤霞, 程浩. 椭圆方程柯西问题磨光正则化参数的后验选取[J]. 山东大学学报(理学版), 2018, 53(2):18-24. DING Fengxia, CHENG Hao. A posteriori choice rule for the mollification regularization parameter for the Cauchy problem of an elliptic equation[J]. Journal of Shandong University(Natural Science), 2018, 53(2):18-24.
[21] QIAN Zhi, FENG Xiaoli. Numerical solution of a 2D inverse heat conduction problem[J]. Inverse Probl Sci Eng, 2013, 21(3):467-484.
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